Rainbow trees in uniformly edge‐colored graphs. Issue 2 (17th June 2022) - Aigner‐Horev, Elad; Hefetz, Dan; Lahiri, Abhiruk
Abstract: We obtain sufficient conditions for the emergence of spanning and almost‐spanning bounded‐degree rainbow trees in various host graphs, having their edges colored independently and uniformly at random, using a predetermined palette. Our first result asserts that a uniform coloring of 𝔾 ( n, ω ( 1 ) / n ), using a palette of size n n , a.a.s. admits a rainbow copy of any given bounded‐degree tree on at most ( 1 − ε ) n (1-ε )n vertices, where ε > 0 ε >0 is arbitrarily small yet fixed. This serves as a rainbow variant of a classical result by Alon et al. pertaining to the embedding of bounded‐degree almost‐spanning prescribed trees in 𝔾 ( n, C / n ), where C > 0 C>0 is independent of n n . Given an n n ‐vertex graph G G with minimum degree at least δ n δ n , where δ > 0 δ >0 is fixed, we use our aforementioned result in order to prove that a uniform coloring of the randomly perturbed graph G ∪ 𝔾 ( n, ω ( 1 ) / n ), using ( 1 + α ) n (1+α )n colors, where α > 0 α >0 is arbitrarily small yet fixed, a.a.s. admits a rainbow copy of any given bounded‐degree spanning tree. This can be viewed as a rainbow variant of a result by Krivelevich et al. who proved that G ∪ 𝔾 ( n, C / n ), where C > 0 C>0 is independent of n n , a.a.s. admits a copy of any given bounded‐degree spanning tree. Finally, and with G G as above, we prove that a uniform coloring of G ∪ 𝔾 ( n, ω ( n − 2 ) ) using n − 1 n-1 colors a.a.s. admits a rainbow spanning tree. Put another way, the trivial lower bound on the size of the palette required for supporting a rainbow spanning tree is also sufficient, essentially as soon as the random perturbation a.a.s. has edges.